An introduction to the general properties of the Fourier series
Introduction
In this module we will discuss the basic properties of the Continuous-Time Fourier Series. We will begin by refreshing your memory of our basic
Fourier series equations:
Let
denote the transformation from
to the Fourier coefficients
maps complex valued functions to sequences of
complex numbers .
The rate of decay of the Fourier series determines if
has
finite energy .
Parsevals theorem demonstration
Interact (when online) with a Mathematica CDF demonstrating Parsevals Theorem. To download, right click and save file as .cdf.
Symmetry properties
Even signals
Even signals
Odd signals
Odd signals
*
Real signals
Real signals
*
*
Differentiation in fourier domain
Since
then
A differentiator
attenuates the low
frequencies in
and
accentuates the high frequencies. It
removes general trends and accentuates areas of sharpvariation.
A common way to mathematically measure the smoothness of a
function
is to see how many derivatives are finite energy.
This is done by looking at the Fourier coefficients of thesignal, specifically how fast they
decay as
.If
and
has the form
,
then
and has the form
.So for the
derivative to have finite energy, we need
thus
decays
faster than
which implies that
or
Thus the decay rate of the Fourier series dictates
smoothness.
Fourier differentiation demonstration
Interact (when online) with a Mathematica CDF demonstrating Differentiation in the Fourier Domain. To download, right click and save file as .cdf.
Integration in the fourier domain
If
then
If
, this expression doesn't make sense.
Integration accentuates low frequencies and attenuates high
frequencies. Integrators bring out the
general
trends in signals and suppress short term variation
(which is noise in many cases). Integrators are
much nicer than differentiators.
Fourier integration demonstration
Interact (when online) with a Mathematica CDF demonstrating Integration in the Fourier Domain. To download, right click and save file as .cdf.
Signal multiplication and convolution
Given a signal
with Fourier coefficients
and a signal
with Fourier coefficients
,
we can define a new signal,
,
where
.
We find that the Fourier Series representation of
,
,
is such that
.
This is to say that signal multiplication in the time domainis equivalent to
signal convolution in the frequency domain, and vice-versa: signal multiplication in the frequency domain is equivalent to signal convolution in the time domain.The proof of this is as follows
Like other Fourier transforms, the CTFS has many useful properties, including linearity, equal energy in the time and frequency domains, and analogs for shifting, differentation, and integration.
Properties of the ctfs
Property
Signal
CTFS
Linearity
Time Shifting
Time Modulation
Multiplication
Continuous Convolution
Questions & Answers
show that the set of all natural number form semi group under the composition of addition
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the
fraction, the value of the fraction becomes 2/3. Find the original fraction.
2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
Solve for the first variable in one of the equations, then substitute the result into the other equation.
Point For:
(6111,4111,−411)(6111,4111,-411)
Equation Form:
x=6111,y=4111,z=−411x=6111,y=4111,z=-411